Almost all integer matrices have no integer eigenvalues

TL;DR

This paper investigates the likelihood that large random integer matrices are singular or have rational eigenvalues, showing these probabilities diminish as the entries' bounds increase, thus almost all such matrices lack integer eigenvalues.

Contribution

It provides new bounds on the probabilities of singularity and rational eigenvalues for random integer matrices, extending previous results to higher dimensions.

Findings

Probability of singularity decreases as k^{-2+ε}

Probability of rational eigenvalues decreases as k^{-1+ε}

Results generalize earlier 2x2 matrix findings

Abstract

For a fixed $n\ge2$, consider an $n\times n$ matrix $M$ whose entries are random integers bounded by $k$ in absolute value. In this paper, we examine the probability that $M$ is singular (hence has eigenvalue 0), and the probability that $M$ has at least one rational eigenvalue. We show that both of these probabilities tend to 0 as $k$ increases. More precisely, we establish an upper bound of size $k^{-2+\epsilon}$ for the probability that $M$ is singular, and size $k^{-1+\epsilon}$ for the probability that $M$ has a rational eigenvalue. These results generalize earlier work by Kowalsky for the case $n=2$ and answer a question posed by Hetzel, Liew, and Morrison.